Tag: wavefunction collapse

• From the Heisenberg cut to the Copenhagen interpretation

  The following post was motivated by this exchange (on X.com), which prompted me to write out my understanding of the Copenhagen interpretation of quantum mechanics and the part the Heisenberg cut plays in it. I haven’t gone into the variants of the interpretation that Maria Violaris brings up; I only focus on understanding what the interpretation does and doesn’t say to begin with, and its history.

  There are many interpretations of what quantum mechanics says about reality. This is unlike classical physics, where theory and reality converge almost perfectly. If using Newton’s laws of motion you determine that a ball flying through the air will have some speed at some point, you’ll find that to be the case when you take measurements. Quantum mechanics on the other hand has some uncertainty baked into the outcomes of certain measurements; there’s no escaping it. That means the mathematical formalism describes only the probability of the outcomes of measurement rather than the event itself, creating a fundamental gap between the theory and observations that different interpretations have tried to bridge with competing philosophical explanations.

  Perhaps the most popular among them is the Copenhagen interpretation: a small 2016 survey found it enjoys the most agreement among physicists; it also holds sway in the popular imagination thanks to Erwin Schrödinger’s thought experiment involving a cat that’s both dead and alive. However, Schrödinger came up with that idea to illustrate his belief that the Copenhagen interpretation of quantum mechanics paints an absurd picture of reality. The interpretation has been refined over time and is more complicated than that, and certainly not absurd.

  In Schrödinger’s thought experiment, the cat is a metaphor for an observable property of a quantum system. That the cat is both dead and alive — a statement that the wavefunction of the property is in a superposition of two (or more) states. When you open the box to see if the cat is dead or alive (but not both) in the metaphor, the description of the system updates from a superposition to a single outcome.

  Note that this is a simplified picture. For a more thoroughgoing account, I recommend Jim Baggott’s post ‘The Copenhagen Confusion’. Here’s a line from the operative passage: “The ‘collapse of the wavefunction’ was never part of the Copenhagen interpretation because the wavefunction isn’t interpreted realistically. The only thing that happens when an electron is detected on a screen in the context of Copenhagen is that we gain knowledge of the position of the electron.” In this post, however, I’m going to flatten these details for simplicity’s sake where necessary.

  

  A useful entry point to the interpretation is the Heisenberg cut, which is a conceptual boundary within the interpretation. It draws the line between the quantum system, i.e. the wavefunction and probabilistic laws, and the measuring apparatus or the observer, described by classical mechanics and deterministic laws. And these two parts of the overall system share a foundational relationship: the Copenhagen interpretation uses this cut to bridge the gap between the mathematical formalism of quantum mechanics and the empirical reality of what scientists observe in a lab.

  In Niels Bohr’s view, the cut is required because humans are macroscopic entities who communicate using classical language. (“It’s very hard to talk quantum using a language originally designed to tell other monkeys where the ripe fruit is”: Terry Pratchett.) Bohr argued that we don’t have a choice but to describe experiments in terms of everyday physics, including positions, momenta, and times, because these concepts also define our cognitive and linguistic capabilities. This means even though the subatomic world is quantum mechanical, the instruments we use to measure it, like photographic plates and our eyes, must be treated as classical objects. The Heisenberg cut is an imaginary boundary in our description of experiments where we stop using quantum concepts and start using classical ones.

  An important feature of the cut is its mobility, i.e. that a person can draw it anywhere in their description of the thought experiment: when a photon of light hits the cat, when a photon reflected by the cat reaches your eye, when you first open the box or somewhere else. According to the Copenhagen interpretation, the physical predictions of quantum mechanics don’t change based on where you make the cut, as long as it is placed somewhere along the chain of measurement. And the cut must exist if you’re to be able to ‘measure’ the system.

  The Heisenberg cut is also intimately tied to the measurement problem. On the quantum side of the cut, the system will evolve according to the Schrödinger equation, which is deterministic and preserves superpositions, i.e. it allows a particle to be in two states at once. On the classical side of the cut, you observe definite outcomes: the particle is either here or there.

  In effect the cut marks the point where multiple possible outcomes give way to a single recorded result. And in the Copenhagen interpretation, this transition isn’t a physical process that can be derived from the Schrödinger equation itself; instead it’s a non-dynamical event that occurs whenever a quantum system interacts with a classical measuring device. This leads to the somewhat paradoxical conclusion that quantum mechanics is a complete theory of the microscopic universe yet it banks on classical concepts (that it can’t make sense of) to make sense of its predictions.

  While both Bohr and Werner Heisenberg, for whom the cut is named, agreed that this cut should exist, they arrived at it for different reasons. Heisenberg treated the cut as a moveable mathematical boundary that separated the object from the subject, highlighting the subjective nature of observation. He was interested in how the observer’s knowledge changed the state of the system. Bohr on the other hand viewed the cut as an epistemological necessity fixed by the experimental arrangement. In other words for Bohr the cut wasn’t about a subjective observer disrupting nature but about the objective impossibility of separating the observer from the observed in the quantum realm (a.k.a. the uncertainty implicit to quantum mechanics).

  Second, let’s look at how the Copenhagen interpretation treats the maths of quantum mechanics. The theory postulates that a quantum system evolves according to the Schrödinger equation. However, our human experience is obviously discontinuous: we see definite outcomes, not superpositions. The ‘collapse’ is the instant when the system switches from its smooth quantum evolution to a single, definite state.

  Without the Heisenberg cut, on the other hand, there’s no logical place for the wavefunction to collapse. If you treated the entire universe — including a subatomic particle, a microscope, a scientist, and the scientist’s brain — as one giant quantum system, everything would just keep evolving according to the Schrödinger equation forever. Eventually you’d end up with a universe in a massive, complex superposition but you’d never arrive at a specific measurement or result. This is actually the premise of the many-worlds interpretation of quantum mechanics, which removes the collapse and thus removes the need for a cut.

  In the Copenhagen interpretation, however, because you eventually arrive at a definite result (and which you need to do for science to be science), you’re forced to draw a line: “Everything on this side is quantum and describes probabilities and everything on that side is classical and describes facts”. The wavefunction ‘collapse’ is defined as the point at which the quantum description gives way to a single, definite experimental outcome. When the quantum system crosses the Heisenberg cut and interacts with the classical side, the wavefunction is said to have collapsed.

  Thus to discuss the Heisenberg cut is essentially to discuss the mechanism of collapse and highlights the implicit dualism of the Copenhagen interpretation: the universe is divided into the observer and the observed. The wavefunction describes what’s being observed and the collapse ensures the observed entity matches the observer’s reality.

  The concept of the cut originated in a few intense months leading up to Heisenberg’s publication of a paper in March 1927. At the time, Heisenberg had been working at Bohr’s institute in Copenhagen on rescuing the concept of particle trajectories, e.g. the tracks of particles recorded in a cloud chamber, which seemed to contradict the (then) new quantum mechanics.

  In 1925, Heisenberg formulated matrix mechanics, the first logically consistent mathematical framework for quantum mechanics. (This invention was an important first step of the ‘new’ quantum mechanics, whose centenary physicists celebrated worldwide last year.) Among other things, matrix mechanics predicted that certain physical quantities, such as energy, take on discrete values. However, this raised questions about reconciling the theory with physicists observing apparently smooth, continuous particle tracks in cloud chambers.

  

  Heisenberg resolved this contradiction by redefining what a ‘path’ actually is in a cloud chamber. This is a device filled with alcohol vapour that’s supersaturated, meaning it’s cooled to the point where it’s just about ready to turn into liquid. When a charged particle moves through this gas, it knocks electrons out of the alcohol molecules, creating a trail of ions. The vapour rapidly turns into liquid droplets around these ions, forming a visible white track that traces the exact path of the subatomic particle through the chamber.

  But Heisenberg argued that we never actually see a continuous path in a cloud chamber — only the sequence of individual droplets formed by ionisation. Solving the problem of the particle’s trajectory in matrix mechanics would never spit out a continuous path but it could determine the probability of an electron’s state transitioning from one discrete droplet to the next.

  When we say an object transitions from point A to point B in everyday life, we mean it moved through the space in between them. But in matrix mechanics, an electron state transitioning between droplets means a discontinuous update of reality rather than movement. In the context of this post, the state of the electron is a mathematical list of properties the electron possesses at the exact moment it hits a gas molecule and creates a droplet.

  So say when it hits droplet 1, the electron has energy E_high, momentum P₁, and is roughly at position X₁. At droplet 2, scientists find the same electron has energy E_low (because it lost some energy when it smashed into the first atom), momentum P₂, and is roughly at position X₂. In Heisenberg’s telling, the laws of physics don’t describe this journey so much as the probability of state 2 happening given state 1 just happened.

  This description resolved Heisenberg’s problem because his maths only handled the energy levels and transitions; it had no variable for the particle’s location at each instant in time. In other words by looking at the cloud chamber and saying, “Aha! This track is just a pile of separate water droplets”, he could claim that the physical world also works like his maths. Which means the path we see in the cloud chamber is just our human brains drawing a line between the dots. The electron itself only becomes classically describable when it hits something.

  In other words, in classical physics, the particle has a path regardless of whether we look at it, and the droplets merely reveal it. In Heisenberg’s view, the particle has no defined position or path in the empty space between the droplets. Instead a path as such comes into view only because the cloud chamber is performing a rapid series of measurements: each droplet represents an observation that forces the electron to take a stand on its position while the eventual smooth line is a mental construct we create by connecting these dots.

  Continuing from this idea, in a famous letter to Wolfgang Pauli and subsequently in his March 1927 paper, The Actual Content of Quantum Theoretical Kinematics and Mechanics, Heisenberg introduced a thought experiment involving a gamma-ray microscope. He argued that to observe an electron, one must hit it with a photon. This interaction would disturb the electron. He initially framed the measurement problem as a physical interaction between the electron (the system) and the photon (the probe), where the act of measurement mechanically disturbed the system.

  Bohr’s critique of Heisenberg’s draft then reforged the cut as a central tenet of the Copenhagen interpretation. When Heisenberg showed Bohr his paper, Bohr tore into it arguing that Heisenberg was wrong to focus on the disturbance because he assumed the electron had a definite position and momentum before the measurement and which the measurement then messed up. Bohr insisted on the more radical view that the properties of the electron aren’t well-defined until the experimental arrangement itself is fixed. For Bohr, the cut wasn’t just where a disturbance happened but the line where the observer switched from using quantum concepts to classical concepts to describe the experiment.

  The conversations on this point between the two men in February and March 1927 were intense, protracted, and emotionally exhausting. Heisenberg was 25 years old at the time and convinced he had solved the riddle of quantum mechanics with his paper whereas Bohr was relentless in his criticism, insisting Heisenberg’s fundamental premise was logically flawed.

  According to historical accounts, including Heisenberg’s own recollections later in life, the discussions would go on for hours, often late into the night. At one point, the combination of mental exhaustion and Bohr’s stubborn refusal to accept Heisenberg’s interpretation caused Heisenberg to break down in tears of frustration. But Heisenberg eventually capitulated, though not entirely: he didn’t rewrite the entire body of his paper but he did add a postscript to the end of the published version where he acknowledged that his explanation of the gamma-ray microscope had been too simplistic and that Bohr’s view regarding the electron’s indefiniteness was the deeper truth.

  The tears were the physical manifestation of the painful process of aligning the two different viewpoints into what became the Copenhagen interpretation. In fact, and at the risk of repetition, let’s treat this interpretation as the peace treaty that reconciled Heisenberg’s idea of uncertainty with Bohr’s idea of complementarity. Heisenberg’s view was initially very mechanical and focused on the observer’s limitations; he held that the fuzziness of the quantum world was a result of our clumsiness: i.e. the reality existed but our clumsy hands destroyed the data every time we tried to touch it. To him the Heisenberg cut was the place where this mechanical disturbance happened.

  Bohr however worked with the concept of complementarity: that the electron has a dual nature, wave and particle, and that these two natures are mutually exclusive, meaning we can’t see both at the same time. And the uncertainty isn’t because we hit the particle but because the electron literally doesn’t have a defined position and momentum at the same time. If you build an experiment to measure its position, the wave nature would vanish, and vice versa. He was saying in effect that the experiment itself defined what reality was allowed to exist at all in that moment.

  The Copenhagen interpretation loosely synthesised these two views, though it leaned heavily toward Bohr’s. It stated that we must accept two contradictory truths: the mathematical formalism (Heisenberg’s matrix mechanics and the Schrödinger equation) that predicts probabilities and the classical world of our measuring devices. The interpretation is the agreement that we can’t speak about what the electron is doing when we aren’t looking. We can only speak about the results of the interaction between the electron and the machine.

  In effect, the Copenhagen interpretation asserts that physics isn’t about the ontological nature of the electron, i.e. what it is, but about the epistemological nature of our knowledge, or what we can say. And the Heisenberg cut is the necessary border where the indefinite, contradictory quantum world based on Bohr’s idea of complementarity is forced to collapse into a single, definite fact.

  If Bohr and Heisenberg provided the philosophical foundation for the Copenhagen interpretation, the Hungarian-American physicist John von Neumann gave it its formal mathematical form in his 1932 book Mathematical Foundations of Quantum Mechanics. Von Neumann was also the one to show that the mathematics of quantum mechanics allowed the cut to be placed anywhere in this chain without changing the final calculated probabilities.

  Where’s Schrödinger’s cat in all of this, then? As it happens, the famous thought experiment in which the cat is both dead and alive is often misunderstood as a quirk of quantum physics; it was actually a scathing piece of satire Schrödinger designed to show that the Copenhagen interpretation was absurd. Schrödinger in fact didn’t believe a cat could be simultaneously dead and alive. His point was that if you followed Bohr and Heisenberg’s logic to its ultimate conclusion, you’d end up with such a nonsensical reality.

  In fact, the thought experiment, published in 1935, targeted the concept of the Heisenberg cut. In the Copenhagen view, a quantum particle like an atom doesn’t have a defined state: it exists in a superposition of all possible states until an observer measures. Schrödinger could accept this for atoms but couldn’t digest the prospect of applying the idea to macroscopic objects.

  In his mental argument, Schrödinger described a radioactive atom placed in a sealed steel box. If the atom decays in a random quantum event, a Geiger counter nearby would push a hammer, which would smash a vial of cyanide and kill a cat. If the atom doesn’t decay, the cat would live. According to the strict logic of the Copenhagen interpretation, this system remains in a superposition until an observer opens the box to check the cat’s existential status. But until the measurement itself, because the atom is both decayed and not decayed, the Geiger counter is both triggered and not triggered, and the cat is simultaneously dead and alive. Schrödinger’s question was about where the quantum ends and the classical world begins. In other words, where’s the Heisenberg cut?

  

  If we make the cut at the Geiger counter, the cat would be a classical object and thus either dead or alive, not both. However, Bohr, Heisenberg, and von Neumann had shown that the cut was mobile. If we moved it to the human observer opening the box, the cat itself would become part of the system’s overall wavefunction — and Schrödinger had contended that treating a living organism as a probability wave was ridiculous. He used the cat to argue that there must be something missing in the theory, some hidden variables or physical reality, that would determine the state of the cat before an observer looks at it.

  For Schrödinger, the cat proved that the Copenhagen interpretation’s refusal to define objective reality between measurements was a philosophical failure. It showed that while the cut could work mathematically, as von Neumann had proved, it led to macroscopic impossibilities in the physical domain.

  The Copenhagen interpretation in turn didn’t surmount Schrödinger’s critique by answering the riddle but by dismissing Schrödinger’s question as unscientific. Bohr argued that Schrödinger was ‘illegally’ extending quantum concepts beyond the point where a classical description would be required. In his view a Geiger counter is a macroscopic measuring device so the cut between the quantum and classical worlds would occur the moment the particle interacts with the Geiger counter. And by the time the signal reaches the hammer, let alone the cat, the quantum description would already have yielded a definite outcome at the measuring device, so the cat would never have had to be described as being in superposition.

  There was also a powerful sociological narrative at the time that painted Schrödinger and Albert Einstein as an ‘old guard’ that was too stuck in classical determinism to accept the radical new truths quantum mechanics was throwing up. By 1935, the Copenhagen interpretation was the dominant orthodoxy among the younger, more productive generation of physicists like Pauli and (to a lesser extent) Paul Dirac, who viewed the cat and the Einstein-Podolsky-Rosen paradox not as genuine physical problems but as the confusion of men who couldn’t let go of the past. The proponents of the interpretation essentially declared that if the theory predicted the results of experiments correctly, then any philosophical discomfort about cats that were both dead and alive was the philosopher’s problem, not the physicist’s. And quantum mechanics perfectly predicted the results of experiments.

  Historical timing also played an important part in cementing the Copenhagen interpretation’s dominance. Shortly after Schrödinger published his paper, physics shifted dramatically from the philosophical debates of the 1920s to the pragmatic urgency of the 1930s and 1940s. The rise of fascism and World War II turned the focus of the community towards nuclear energy and The Bomb. In this environment, the “shut up and calculate” approach — a phrase coined later to describe this attitude — took over and physicists shelved questions about the reality of the cat as irrelevant metaphysics.

  The interpretation was also shielded by von Neumann’s mathematical authority. His 1932 book also claimed to show that ‘hidden variable’ theories, i.e. which would restore a specific reality to the cat independent of observation, were mathematically impossible. While Grete Hermann and John Bell later found this proof to be circular, for decades it served as a brick wall that convinced the physics community that there was literally no alternative to the Copenhagen interpretation.

  2026.01.31

• Dispelling Maxwell’s demon

  Maxwell’s demon is one of the most famous thought experiments in the history of physics, a puzzle first posed in the 1860s that continues to shape scientific debates to this day. I’ve struggled to make sense of it for years. Last week I had some time and decided to hunker down and figure it out, and I think I succeeded. The following post describes the fruits of my efforts.

  At first sight, the Maxwell’s demon paradox seems odd because it presents a supernatural creature tampering with molecules of gas. But if you pare down the imagery and focus on the technological backdrop of the time of James Clerk Maxwell, who proposed it, a profoundly insightful probe of the second law of thermodynamics comes into view.

  The thought experiment asks a simple question: if you had a way to measure and control molecules with perfect precision and at no cost, will you able to make heat flow backwards, as if in an engine?

  Picture a box of air divided into two halves by a partition. In the partition is a very small trapdoor. It has a hinge so it can swing open and shut. Now imagine a microscopic valve operator that can detect the speed of each gas molecule as it approaches the trapdoor, decide whether to open or close the door, and actuate the door accordingly.

  The operator follows two simple rules: let fast molecules through from left to right and let slow molecules through from right to left. The temperature of a system is nothing but the average kinetic energy of its constituent particles. As the operator operates, over time the right side will heat up and the left side will cool down — thus producing a temperature gradient for free. Where there’s a temperature gradient, it’s possible to run a heat engine. (The internal combustion engine in fossil-fuel vehicles is a common example.)

  

  But the possibility that this operator can detect and sort the molecules, thus creating the temperature gradient without consuming some energy of its own, seems to break the second law of thermodynamics. The second law states that the entropy of a closed system increases over time — whereas the operator ensures that the temperature will decrease, violating the law. This was the Maxwell’s demon thought experiment, with the demon as a whimsical stand-in for the operator.

  The paradox was made compelling by the silent assumption that the act of sorting the molecules could have no cost — i.e. that the imagined operator didn’t add energy to the system (the air in the box) but simply allowed molecules that are already in motion to pass one way and not the other. In this sense the operator acted like a valve or a one-way gate. Devices of this kind — including check valves, ratchets, and centrifugal governors — were already familiar in the 19th century. And scientists assumed that if they were scaled down to the molecular level, they’d be able to work without friction and thus separate hot and cold particles without drawing more energy to overcome that friction.

  This detail is in fact the fulcrum of the paradox, and the thing that’d kept me all these years from actually understanding what the issue was. Maxwell et al. assumed that it was possible that an entity like this gate could exist: one that, without spending energy to do work (and thus increase entropy), could passively, effortlessly sort the molecules. Overall, the paradox stated that if such a sorting exercise really had no cost, the second law of thermodynamics would be violated.

  The second law had been established only a few decades before Maxwell thought up this paradox. If entropy is taken to be a measure of disorder, the second law states that if a system is left to itself, heat will not spontaneously flow from cold to hot and whatever useful energy it holds will inevitably degrade into the random motion of its constituent particles. The second law is the reason why perpetual motion machines are impossible, why the engines in our cars and bikes can’t be 100% efficient, and why time flows in one specific direction (from past to future).

  Yet Maxwell’s imagined operator seemed to be able to make heat flow backwards, sifting molecules so that order increases spontaneously. For many decades, this possibility challenged what physicists thought they knew about physics. While some brushed it off as a curiosity, others contended that the demon itself must expend some energy to operate the door and that this expense would restore the balance. However, Maxwell had been careful when he conceived the thought experiment: he specified that the trapdoor was small and moved without friction, so it could in principle operate in a negligible way. The real puzzle lay elsewhere.

  In 1929, the Hungarian physicist Leó Szilard sharpened the problem by boiling it down to a single-particle machine. This so-called Szilard engine imagined one gas molecule in a box with a partition that could be inserted or removed. By observing on which side the molecule lay and then allowing it to push a piston, the operator could apparently extract work from a single particle at uniform temperature. Szilard showed that the key step was not the movement of the piston but the acquisition of information: knowing where the particle was. That is, Szilard reframed the paradox to be not about the molecules being sorted but about an observer making a measurement.

  (Aside: Szilard was played by Máté Haumann in the 2023 film Oppenheimer.)

  

  The next clue to cracking the puzzle came in the mid-20th century from the growing field of information theory. In 1961, the German-American physicist Rolf Landauer proposed a principle that connected information and entropy directly. Landauer’s principle states that while it’s possible in principle to acquire information in a reversible way — i.e. to be able to acquire it as well as lose it — erasing information from a device with memory has a non-zero thermodynamic cost that can’t be avoided. That is, the act of resetting a memory register of one bit to a standard state generates a small amount of entropy (proportional to Boltzmann’s constant multiplied by the logarithm of two).

  The American information theorist Charles H. Bennett later built on Landauer’s principle and argued that Maxwell’s demon could gather information and act on it — but in order to continue indefinitely, it’d have to erase or overwrite its memory. And that this act of resetting would generate exactly the entropy needed to compensate for the apparent decrease, ultimately preserving the second law of thermodynamics.

  Taken together, Maxwell’s demon was defeated not by the mechanics of the trapdoor but by the thermodynamic cost of processing information. Specifically, the decrease in entropy as a result of the molecules being sorted by their speed is compensated for by the increase in entropy due to the operator’s rewriting or erasure of information about the molecules’ speed. Thus a paradox that’d begun as a challenge to thermodynamics ended up enriching it — by showing information could be physical. It also revealed to scientists that entropy is disorder in matter and energy as well as is linked to uncertainty and information.

  Over time, Maxwell’s demon also became a fount of insight across multiple branches of physics. In classical thermodynamics, for example, entropy came to represent a measure of the probabilities that the system could exist in different combinations of microscopic states. That is, the probabilities referred to the likelihood that a given set of molecules could be arranged in one way instead of another. In statistical mechanics, Maxwell’s demon gave scientists a concrete way to think about fluctuations. In any small system, random fluctuations can reduce entropy for some time in a small portion. While the demon seemed to exploit these fluctuations, the laws of probability were found to ensure that on average, entropy would increase. So the demon became a metaphor for how selection based on microscopic knowledge could alter outcomes but also why such selection can’t be performed without paying a cost.

  

  For information theorists and computer scientists, the demon was an early symbol of the deep ties between computation and thermodynamics. Landauer’s principle showed that erasing information imposes a minimum entropy cost — an insight that matters for how computer hardware should be designed. The principle also influenced debates about reversible computing, where the goal is to design logic gates that don’t ever erase information and thus approach zero energy dissipation. In other words, Maxwell’s demon foreshadowed modern questions about how energy-efficient computing could really be.

  Even beyond physics, the demon has seeped into philosophy, biology, and social thought as a symbol of control and knowledge. In biology, the resemblance between the demon and enzymes that sorts molecules has inspired metaphors about how life maintains order. In economics and social theory, the demon has been used to discuss the limits of surveillance and control. The lesson has been the same in every instance: that information is never free and that the act of using it imposes inescapable energy costs.

  I’m particularly taken by the philosophy that animates the paradox. Maxwell’s demon was introduced as a way to dramatise the tension between the microscopic reversibility of physical laws and the macroscopic irreversibility encoded in the second law of thermodynamics. I found that a few questions in particular — whether the entropy increase due to the use of information is a matter of an observer’s ignorance (i.e. because the observer doesn’t know which particular microstate the system occupies at any given moment), whether information has physical significance, and whether the laws of nature really guarantee the irreversibility we observe — have become touchstones in the philosophy of physics.

  In the mid-20th century, the Szilard engine became the focus of these debates because it refocused the second law from molecular dynamics to the cost of acquiring information. Later figures such as the French physicist Léon Brillouin and the Hungarian-Canadian physicist Dennis Gabor claimed that it’s impossible to measure something without spending energy. Critics however countered that these requirements stipulated the need for specific technologies that would in turn smuggle in some limitations — rather than stipulate the presence of a fundamental principle. That is to say, the debate among philosophers became whether Maxwell’s demon was prevented from breaking the second law by deep and hitherto hidden principles or by engineering challenges.

  This gridlock was broken when physicists observed that even a demon-free machine must leave some physical trace of its interactions with the molecule. That is, any device that sorts particles will end up in different physical states depending on the outcome, and to complete a thermodynamic cycle those states must be reset. Here, the entropy is not due to the informational content but due to the logical structure of memory. Landauer solidified this with his principle that logically irreversible operations such as erasure carry a minimum thermodynamic cost. Bennett extended this by saying that measurements can be made reversibly but not erasure. The philosophical meaning of both these arguments is that entropy increase isn’t just about ignorance but also about parts of information processing being irreversible.

  

  In the quantum domain, the philosophical puzzles became more intense. When an object is measured in quantum mechanics, it isn’t just about an observer updating the information they have about the object — the act of measuring also seems to alter the object’s quantum states. For example, in the Schrödinger’s cat thought experiment, checking whether there’s a cat in the box also causes the cat to default to one of two states: dead or alive. Quantum physicists have recreated Maxwell’s demon in new ways in order to check whether the second law continues to hold. And over the course of many experiments, they’ve concluded that indeed it does.

  The second law didn’t break even when Maxwell’s demon could exploit phenomena that aren’t available in the classical domain, including quantum entanglement, superposition, and tunnelling. This was because, among others, quantum mechanics also has some restrictive rules of its own. For one, some physicists have tried to design “quantum demons” that use quantum entanglement between particles to sort them without expending energy. But these experiments have found that as soon as the demon tries to reset its memory and start again, it must erase the record of what happened before. This step destroys the advantage and the entropy cost returns. The overall result is that even a “quantum demon” gains nothing in the long run.

  For another, the no-cloning theorem states that you can’t make a perfect copy of an unknown quantum state. If the demon could freely copy every quantum particle it measured, it could retain flawless records while still resetting its memory, this avoiding the usual entropy cost. The theorem blocks this strategy by forbidding perfect duplication, ensuring that information can’t be ‘multiplied’ without limit. Similarly, the principle of unitarity implies that a system will always evolve in a way that preserves overall probabilities. As a result, quantum phenomena can’t selectively amplify certain outcomes while discarding others. For the demon, this means it can’t secretly limit the range of possible states the system can occupy into a smaller set where the system has lower entropy, because unitarity guarantees that the full spread of possibilities is preserved across time.

  All these rules together prevent the demon from multiplying or rearranging quantum states in a way that would allow it to beat the second law.

  Then again, these ‘blocks’ that prevent Maxwell’s demon from breaking the second law of thermodynamics in the quantum realm raise a puzzle of their own: is the second law of thermodynamics guaranteed no matter how we interpret quantum mechanics? ‘Interpreting quantum mechanics’ means to interpret what the rules of quantum mechanics say about reality, a topic I covered at length in a recent post. Some interpretations say that when we measure a quantum system, its wavefunction “collapses” to a definite outcome. Others say collapse never happens and that measurement is just entangled with the environment, a process called decoherence. The Maxwell’s demon thought experiment thus forces the question: is the second law of thermodynamics safe in a particular interpretation of quantum mechanics or in all interpretations?

  

  Landauer’s idea, that erasing information always carries a cost, also applies to quantum information. Even if Maxwell’s demon used qubits instead of bits, it won’t be able to escape the fact that to reuse its memory, it must erase the record, which will generate heat. But then the question becomes more subtle in quantum systems because qubits can be entangled with each other, and their delicate coherence — the special quantum link between quantum states — can be lost when information is processed. This means scientists need to carefully separate two different ideas of entropy: one based on what we as observers don’t know (our ignorance) and another based on what the quantum system itself has physically lost (by losing coherence).

  The lesson is that the second law of thermodynamics doesn’t just guard the flow of energy. In the quantum realm it also governs the flow of information. Entropy increases not only because we lose track of details but also because the very act of erasing and resetting information, whether classical or quantum, forces a cost that no demon can avoid.

  Then again, some philosophers and physicists have resisted the move to information altogether, arguing that ordinary statistical mechanics suffices to resolve the paradox. They’ve argued that any device designed to exploit fluctuations will be subject to its own fluctuations, and thus in aggregate no violation will have occurred. In this view, the second law is self-sufficient and doesn’t need the language of information, memory or knowledge to justify itself. This line of thought is attractive to those wary of anthropomorphising physics even if it also risks trivialising the demon. After all, the demon was designed to expose the gap between microscopic reversibility and macroscopic irreversibility, and simply declaring that “the averages work out” seems to bypass the conceptual tension.

  Thus, the philosophical significance of Maxwell’s demon is that it forces us to clarify the nature of entropy and the second law. Is entropy tied to our knowledge/ignorance of microstates, or is it ontic, tied to the irreversibility of information processing and computation? If Landauer is right, handling information and conserving energy are ‘equally’ fundamental physical concepts. If the statistical purists are right, on the other hand, then information adds nothing to the physics and the demon was never a serious challenge. Quantum theory can further stir both pots by suggesting that entropy is closely linked to the act of measurement, of quantum entanglement, and how quantum systems ‘collapse’ to classical ones by the process of decoherence. The demon debate therefore tests whether information is a physically primitive entity or a knowledge-based tool. Either way, however, Maxwell’s demon endures as a parable.

  Ultimately, what makes Maxwell’s demon a gift that keeps giving is that it works on several levels. On the surface it’s a riddle about sorting molecules between two chambers. Dig a little deeper and it becomes a probe into the meaning of entropy. If you dig even further, it seems to be a bridge between matter and information. As the Schrödinger’s cat thought experiment dramatised the oddness of quantum superposition, Maxwell’s demon dramatised the subtleties of thermodynamics by invoking a fantastical entity. And while Schrödinger’s cat forces us to ask what it means for a macroscopic system to be in two states at once, Maxwell’s demon forces us to ask what it means to know something about a system and whether that knowledge can be used without consequence.

  2025.09.19

• What on earth is a wavefunction?

  If you drop a pebble into a pond, ripples spread outward in gentle circles. We all know this sight, and it feels natural to call them waves. Now imagine being told that everything — from an electron to an atom to a speck of dust — can also behave like a wave, even though they are made of matter and not water or air. That is the bold claim of quantum mechanics. The waves in this case are not ripples in a material substance. Instead, they are mathematical entities known as wavefunctions.

  At first, this sounds like nothing more than fancy maths. But the wavefunction is central to how the quantum world works. It carries the information that tells us where a particle might be found, what momentum it might have, and how it might interact. In place of neat certainties, the quantum world offers a blur of possibilities. The wavefunction is the map of that blur. The peculiar thing is, experiments show that this ‘blur’ behaves as though it is real. Electrons fired through two slits make interference patterns as though each one went through both slits at once. Molecules too large to see under a microscope can act the same way, spreading out in space like waves until they are detected.

  So what exactly is a wavefunction, and how should we think about it? That question has haunted physicists since the early 20th century and it remains unsettled to this day.

  In classical life, you can say with confidence, “The cricket ball is here, moving at this speed.” If you can’t measure it, that’s your problem, not nature’s. In quantum mechanics, it is not so simple. Until a measurement is made, a particle does not have a definite position in the classical sense. Instead, the wavefunction stretches out and describes a range of possibilities. If the wavefunction is sharply peaked, the particle is most likely near a particular spot. If it is wide, the particle is spread out. Squaring the wavefunction’s magnitude gives the probability distribution you would see in many repeated experiments.

  If this sounds abstract, remember that the predictions are tangible. Interference patterns, tunnelling, superpositions, entanglement — all of these quantum phenomena flow from the properties of the wavefunction. It is the script that the universe seems to follow at its smallest scales.

  To make sense of this, many physicists use analogies. Some compare the wavefunction to a musical chord. A chord is not just one note but several at once. When you play it, the sound is rich and full. Similarly, a particle’s wavefunction contains many possible positions (or momenta) simultaneously. Only when you press down with measurement do you “pick out” a single note from the chord.

  Others have compared it to a weather forecast. Meteorologists don’t say, “It will rain here at exactly 3:07 pm.” They say, “There’s a 60% chance of showers in this region.” The wavefunction is like nature’s own forecast, except it is more fundamental: it is not our ignorance that makes it probabilistic, but the way the universe itself behaves.

  Mathematically, the wavefunction is found by solving the Schrödinger equation, which is a central law of quantum physics. This equation describes how the wavefunction changes in time. It is to quantum mechanics what Newton’s second law (F = ma) is to classical mechanics. But unlike Newton’s law, which predicts a single trajectory, the Schrödinger equation predicts the evolving shape of probabilities. For example, it can show how a sharply localised wavefunction naturally spreads over time, just like a drop of ink disperses in water. The difference is that the spreading is not caused by random mixing but by the fundamental rules of the quantum world.

  But does that mean the wavefunction is real, like a water wave you can touch, or is it just a clever mathematical fiction?

  There are two broad camps. One camp, sometimes called the instrumentalists, argues the wavefunction is only a tool for making predictions. In this view, nothing actually waves in space. The particle is simply somewhere, and the wavefunction is our best way to calculate the odds of finding it. When we measure, we discover the position, and the wavefunction ‘collapses’ because our information has been updated, not because the world itself has changed.

  The other camp, the realists, argues that the wavefunction is as real as any energy field. If the mathematics says a particle is spread out across two slits, then until you measure it, the particle really is spread out, occupying both paths in a superposed state. Measurement then forces the possibilities into a single outcome, but before that moment, the wavefunction’s broad reach isn’t just bookkeeping: it’s physical.

  This isn’t an idle philosophical spat. It has consequences for how we interpret famous paradoxes like Schrödinger’s cat — supposedly “alive and dead at once until observed” — and for how we understand the limits of quantum mechanics itself. If the wavefunction is real, then perhaps macroscopic objects like cats, tables or even ourselves can exist in superpositions in the right conditions. If it is not real, then quantum mechanics is only a calculating device, and the world remains classical at larger scales.

  The ability of a wavefunction to remain spread out is tied to what physicists call coherence. A coherent state is one where the different parts of the wavefunction stay in step with each other, like musicians in an orchestra keeping perfect time. If even a few instruments go off-beat, the harmony collapses into noise. In the same way, when coherence is lost, the wavefunction’s delicate correlations vanish.

  Physicists measure this ‘togetherness’ with a parameter called the coherence length. You can think of it as the distance over which the wavefunction’s rhythm remains intact. A laser pointer offers a good everyday example: its light is coherent, so the waves line up across long distances, allowing a sharp red dot to appear even all the way across a lecture hall. By contrast, the light from a torch is incoherent: the waves quickly fall out of step, producing only a fuzzy glow. In the quantum world, a longer coherence length means the particle’s wavefunction can stay spread out and in tune across a larger stretch of space, making the object more thoroughly delocalised.

  However, coherence is fragile. The world outside — the air, the light, the random hustle of molecules — constantly disturbs the system. Each poke causes the system to ‘leak’ information, collapsing the wavefunction’s delicate superposition. This process is called decoherence, and it explains why we don’t see cats or chairs spread out in superpositions in daily life. The environment ‘measures’ them constantly, destroying their quantum fuzziness.

  One frontier of modern physics is to see how far coherence can be pushed before decoherence wins. For electrons and atoms, the answer is “very far”. Physicists have found their wavefunctions can stretch across micrometres or more. They have also demonstrated coherence with molecules with thousands of atoms, but keeping them coherent has been much more difficult. For larger solid objects, it’s harder still.

  Physicists often talk about expanding a wavefunction. What they mean is deliberately increasing the spatial extent of the quantum state, making the fuzziness spread wider, while still keeping it coherent. Imagine a violin string: if it vibrates softly, the motion is narrow; if it vibrates with larger amplitude, it spreads. In quantum mechanics, expansion is more subtle but the analogy holds: you want the wavefunction to cover more ground not through noise or randomness but through genuine quantum uncertainty.

  Another way to picture it is as a drop of ink released into clear water. At first, the drop is tight and dark. Over time, it spreads outward, thinning and covering more space. Expanding a quantum wavefunction is like speeding up this spreading process, but with a twist: the cloud must remain coherent. The ink can’t become blotchy or disturbed by outside currents. Instead, it must preserve its smooth, wave-like character, where all parts of the spread remain correlated.

  How can this be done? One way is to relax the trap that’s being used to hold the particle in place. In physics, the trap is described by a potential, which is just a way of talking about how strong the forces are that pull the particle back towards the centre. Imagine a ball sitting in a bowl. The shape of the bowl represents the potential. A deep, steep bowl means strong restoring forces, which prevent the ball from moving around. A shallow bowl means the forces are weaker. That is, if you suddenly make the bowl shallower, the ball is less tightly confined and can explore more space. In the quantum picture, reducing the stiffness of the potential is like flattening the bowl, which allows the wavefunction to swell outward. If you later return the bowl to its steep form, you can catch the now-broader state and measure its properties.

  The challenge is to do this fast and cleanly, before decoherence destroys the quantum character. And you must measure in ways that reveal quantum behaviour rather than just classical blur.

  This brings us to an experiment reported on August 19 in Physical Review Letters, conducted by researchers at ETH Zürich and their collaborators. It seems the researchers have achieved something unprecedented: they prepared a small silica sphere, only about 100 nm across, in a nearly pure quantum state and then expanded its wavefunction beyond the natural zero-point limit. This means they coherently stretched the particle’s quantum fuzziness farther than the smallest quantum wiggle that nature usually allows, while still keeping the state coherent.

  To appreciate why this matters, let’s consider the numbers. The zero-point motion of their nanoparticle — the smallest possible movement even at absolute zero — is about 17 picometres (one picometre is a trillionth of a meter). Before expansion, the coherence length was about 21 pm. After the expansion protocol, it reached roughly 73 pm, more than tripling the initial reach and surpassing the ground-state value. For something as massive as a nanoparticle, this is a big step.

  The team began by levitating a silica nanoparticle in an optical tweezer, created by a tightly focused laser beam. The particle floated in an ultra-high vacuum at a temperature of just 7 K (-266º C). These conditions reduced outside disturbances to almost nothing.

  Next, they cooled the particle’s motion close to its ground state using feedback control. By monitoring its position and applying gentle electrical forces through the surrounding electrodes, they damped its jostling until only a fraction of a quantum of motion remained. At this point, the particle was quiet enough for quantum effects to dominate.

  The core step was the two-pulse expansion protocol. First, the researchers switched off the cooling and briefly lowered the trap’s stiffness by reducing the laser power. This allowed the wavefunction to spread. Then, after a carefully timed delay, they applied a second softening pulse. This sequence cancelled out unwanted drifts caused by stray forces while letting the wavefunction expand even further.

  Finally, they restored the trap to full strength and measured the particle’s motion by studying how they scattered light. Repeating this process hundreds of times gave them a statistical view of the expanded state.

  The results showed that the nanoparticle’s wavefunction expanded far beyond its zero-point motion while still remaining coherent. The coherence length grew more than threefold, reaching 73 ± 34 pm. Per the team, this wasn’t just noisy spread but genuine quantum delocalisation.

  More strikingly, the momentum of the nanoparticle had become ‘squeezed’ below its zero-point value. In other words, while uncertainty over the particle’s position increased, that over its momentum decreased, in keeping with Heisenberg’s uncertainty principle. This kind of squeezed state is useful because it’s especially sensitive to feeble external forces.

  The data matched theoretical models that considered photon recoil to be the main source of decoherence. Each scattered photon gave the nanoparticle a small kick, and this set a fundamental limit. The experiment confirmed that photon recoil was indeed the bottleneck, not hidden technical noise. The researchers have suggested using dark traps in future — trapping methods that use less light, such as radio-frequency fields — to reduce this recoil. With such tools, the coherence lengths can potentially be expanded to scales comparable to the particle’s size. Imagine a nanoparticle existing in a state that spans its own diameter. That would be a true macroscopic quantum object.

  This new study pushes quantum mechanics into a new regime. Thus far, large, solid objects like nanoparticles could be cooled and controlled, but their coherence lengths stayed pinned near the zero-point level. Here, the researchers were able to deliberately increase the coherence length beyond that limit, and in doing so showed that quantum fuzziness can be engineered, not just preserved.

  The implications are broad. On the practical side, delocalised nanoparticles could become extremely sensitive force sensors, able to detect faint electric or gravitational forces. On the fundamental side, the ability to hold large objects in coherent, expanded states is a step towards probing whether gravity itself has quantum features. Several theoretical proposals suggest that if two massive objects in superposition can become entangled through their mutual gravity, it would prove gravity must be quantum. To reach that stage, experiments must first learn to create and control delocalised states like this one.

  The possibilities for sensing in particular are exciting. Imagine a nanoparticle prepared in a squeezed, delocalised state being used to detect the tug of an unseen mass nearby or to measure an electric field too weak for ordinary instruments. Some physicists have speculated that such systems could help search for exotic particles such as certain dark matter candidates, which might nudge the nanoparticle ever so slightly. The extreme sensitivity arises because a delocalised quantum object is like a feather balanced on a pin: the tiniest push shifts it in measurable ways.

  There are also parallels with past breakthroughs. The Laser Interferometer Gravitational-wave Observatories, which detect gravitational waves, rely on manipulating quantum noise in light to reach unprecedented sensitivity. The ETH Zürich experiment has extended the same philosophy into the mechanical world of nanoparticles. Both cases show that pushing deeper into quantum control could yield technologies that were once unimaginable.

  But beyond the technologies also lies a more interesting philosophical edge. The experiment strengthens the case that the wavefunction behaves like something real. If it were only an abstract formula, could we stretch it, squeeze it, and measure the changes in line with theory? The fact that researchers can engineer the wavefunction of a many-atom object and watch it respond like a physical entity tilts the balance towards reality. At the least, it shows that the wavefunction is not just a mathematical ghost. It’s a structure that researchers can shape with lasers and measure with detectors.

  There are also of course the broader human questions. If nature at its core is described not by certainties but by probabilities, then philosophers must rethink determinism, the idea that everything is fixed in advance. Our everyday world looks predictable only because decoherence hides the fuzziness. But under carefully controlled conditions, that fuzziness comes back into view. Experiments like this remind us that the universe is stranger, and more flexible, than classical common sense would suggest.

  The experiment also reminds us that the line between the quantum and classical worlds is not a brick wall but a veil — thin, fragile, and possibly removable in the right conditions. And each time we lift it a little further, we don’t just see strange behaviour: we also glimpse sensors more sensitive than ever, tests of gravity’s quantum nature, and perhaps someday, direct encounters with macroscopic superpositions that will force us to rewrite what we mean by reality.

  2025.08.23

• A latent monadology: An extended revisitation of the mind-body problem

  

  In an earlier post, I’d spoken about a certain class of mind-body interfacing problems (the way I’d identified it): evolution being a continuous process, can psychological changes effected in a certain class of people identified solely by cultural practices “spill over” as modifications of evolutionary goals? There were some interesting comments on the post, too. You may read them here.

  However, the doubt was only the latest in a series of others like it. My interest in the subject was born with a paper I’d read quite a while ago that discussed two methods either of which humankind could possibly use to recreate the human brain as a machine. The first method, rather complexly laid down, was nothing but the ubiquitous recourse called reverse-engineering. Study the brain, understand what it’s made of, reverse all known cause-effect relationships associated with the organ, then attempt to recreate the cause using the effect in a laboratory with suitable materials to replace the original constituents.

  The second method was much more interesting (this bias could explain the choice of words in the previous paragraph). Essentially, it described the construction of a machine that could perform all the known functions of the brain. Then, this machine would have to be subjected to a learning process, through which it would acquire new skills while it retained and used the skills it’s already been endowed with. After some time, if the learnt skills, so chosen to reflect real human skills, are deployed by the machine to recreate human endeavor, then the machine is the brain.

  Why I like this method better than the reverse-engineered brain is because it takes into account the ability to learn as a function of the brain, resulting in a more dynamic product. The notion of the brain as a static body is definitively meaningless as, axiomatically, conceiving of it as a really powerful processor stops short of such Leibnizian monads as awareness and imagination. While these two “entities” evade comprehension, subtracting the ability to, yes, somehow recreate them doesn’t yield a convincing brain as it is. And this is where I believe the mind-body problem finds solution. For the sake of argument, let’s discuss the issue differentially.

  

  Hold as constant: Awareness
  Hold as variable: Imagination

  The brain is aware, has been aware, must be aware in the future. It is aware of the body, of the universe, of itself. In order to be able to imagine, therefore, it must concurrently trigger, receive, and manipulate different memorial stimuli to construct different situations, analyze them, and arrive at a conclusion about different operational possibilities in each situation. Note: this process is predicated on the inability of the brain to birth entirely original ideas, an extension of the fact that a sleeping person cannot be dreaming of something he has not interacted with in some way.

  Hold as constant: Imagination
  Hold as variable: Awareness

  At this point, I need only prove that the brain can arrive at an awareness of itself, the body, and the universe, through a series of imaginative constructs, in order to hold my axiom as such. So, I’m going to assume that awareness came before imagination did. This leaves open the possibility that with some awareness, the human mind is able to come up with new ways to parse future stimuli, thereby facilitating understanding and increasing the sort of awareness of everything that better suits one’s needs and environment.

  Now, let’s talk about the process of learning and how it sits with awareness, imagination, and consciousness, too. This is where I’d like to introduce the metaphor called Leibniz’s gap. In 1714, Gottfried Leibniz’s ‘Principes de la Nature et de la Grace fondés en raison‘ was published in the Netherlands. In the work, which would form the basis of modern analytic philosophy, the philosopher-mathematician argues that there can be no physical processes that can be recorded or tracked in any way that would point to corresponding changes in psychological processes.

  … supposing that there were a mechanism so constructed as to think, feel and have perception, we might enter it as into a mill. And this granted, we should only find on visiting it, pieces which push one against another, but never anything by which to explain a perception. This must be sought, therefore, in the simple substance, and not in the composite or in the machine.

  If any technique was found that could span the distance between these two concepts – the physical and the psychological – then Leibniz says the technique will effectively bridge Leibniz’s gap: the symbolic distance between the mind and the body.

  Now it must be remembered that the German was one of the three greatest, and most fundamentalist, rationalists of the 17th century: the other two were Rene Descartes and Baruch Spinoza (L-D-S). More specifically: All three believed that reality was composed fully of phenomena that could be explained by applying principles of logic to a priori, or fundamental, knowledge, subsequently discarding empirical evidence. If you think about it, this approach is flawless: if the basis of a hypothesis is logical, and if all the processes of development and experimentation on it are founded in logic, then the conclusion must also be logical.

  

  However, where this model does fall short is in describing an anomalous phenomenon that is demonstrably logical but otherwise inexplicable in terms of the dominant logical framework. This is akin to Thomas Kuhn’s philosophy of science: a revolution is necessitated when enough anomalies accumulate that defy the reign of an existing paradigm, but until then, the paradigm will deny the inclusion of any new relationships between existing bits of data that don’t conform to its principles.

  When studying the brain (and when trying to recreate it in a lab), Leibniz’s gap, as understood by L-D-S, cannot be applied for various reasons. First: the rationalist approach doesn’t work because, while we’re seeking logical conclusions that evolve from logical starts, we’re in a good position to easily disregard the phenomenon called emergence that is prevalent in all simple systems that have high multiplicity. In fact, ironically, the L-D-S approach might be more suited for grounding empirical observations in logical formulae because it is only then that we run no risk of avoiding emergent paradigms.

  

  Second: It is important to not disregard that humans do not know much about the brain. As elucidated in the less favored of the two-methods I’ve described above, were we to reverse-engineer the brain, we can still only make the new-brain do what we already know that it already does. The L-D-S approach takes complete knowledge of the brain for granted, and works post hoc ergo propter hoc (“correlation equals causation”) to explain it.

  Therefore, in order to understand the brain outside the ambit of rationalism (but still definitely within the ambit of empiricism), introspection need not be the only way. We don’t always have to scrutinize our thoughts to understand how we assimilated them in the first place, and then move on from there, when we can think of the brain itself as the organ bridging Leibniz’s gap. At this juncture, I’d like to reintroduce the importance of learning as a function of the brain.

  To think of the brain as residing at a nexus, the most helpful logical frameworks are the computational theory of the mind (CTM) and the Copenhagen interpretation of quantum mechanics (QM).

  

  In the CTM-framework, the brain is a processor, and the mind is the program that it’s running. Accordingly, the organ works on a set of logical inputs, each of which is necessarily deterministic and non-semantic; the output, by extension, is the consequence of an algorithm, and each step of the algorithm is a mental state. These mental states are thought to be more occurrent than dispositional, i.e., more tractable and measurable than the psychological emergence that they effect. This is the break from Leibniz’s gap that I was looking for.

  Because the inputs are non-semantic, i.e., interpreted with no regard for what they mean, it doesn’t mean the brain is incapable of processing meaning or conceiving of it in any other way in the CTM-framework. The solution is a technical notion called formalization, which the Stanford Encyclopedia of Philosophy describes thus:

  … formalization shows us how semantic properties of symbols can (sometimes) be encoded in syntactically-based derivation rules, allowing for the possibility of inferences that respect semantic value to be carried out in a fashion that is sensitive only to the syntax, and bypassing the need for the reasoner to have employ semantic intuitions. In short, formalization shows us how to tie semantics to syntax.

  A corresponding theory of networks that goes with such a philosophy of the brain is connectionism. It was developed by Walter Pitts and Warren McCulloch in 1943, and subsequently popularized by Frank Rosenblatt (in his 1957 conceptualization of the Perceptron, a simplest feedforward neural network), and James McClelland and David Rumelhart (‘Learning the past tenses of English verbs: Implicit rules or par­allel distributed processing’, In B. MacWhinney (Ed.), Mechanisms of Language Acquisition (pp. 194-248). Mah­wah, NJ: Erlbaum) in 1987.

  

  As described, the L-D-S rationalist contention was that fundamental entities, or monads or entelechies, couldn’t be looked for in terms of physiological changes in brain tissue but in terms of psychological manifestations. The CTM, while it didn’t set out to contest this, does provide a tensor in which the inputs and outputs are correlated consistently through an algorithm with a neural network for an architecture and a Turing/Church machine for an algorithmic process. Moreover, this framework’s insistence on occurrent processes is not the defier of Leibniz: the occurrence is presented as antithetical to the dispositional.

  

  The defier of Leibniz is the CTM itself: if all of the brain’s workings can be elucidated in terms of an algorithm, inputs, a formalization module, and outputs, then there is no necessity to suppress any thoughts to the purely-introspectionist level (The domain of CTM, interestingly, ranges all the way from the infraconscious to the set of all modular mental processes; global mental processes, as described by Jerry Fodor in 2000, are excluded, however).

  Where does quantum mechanics (QM) come in, then? Good question. The brain is a processor. The mind is a program. The architecture is a neural network. The process is that of a Turing machine. But how is the information between received and transmitted? Since we were speaking of QM, more specifically the Copenhagen interpretation of it, I suppose it’s obvious that I’m talking about electrons and electronic and electrochemical signals being transmitted through sensory, motor, and interneurons. While we’re assuming that the brain is definable by a specific processual framework, we still don’t know if the interaction between the algorithm and the information is classical or quantum.

  While the classical outlook is more favorable because almost all other parts of the body are fully understand in terms of classical biology, there could be quantum mechanical forces at work in the brain because – as I’ve said before – we’re in no way to confirm or deny if it’s purely classical or purely non-classical operationally. However, assuming that QM is at work, then associated aspects of the mind, such as awareness, consciousness, and imagination, can be described by quantum mechanical notions such as the wavefunction-collapse and Heisenberg’s uncertainty principle – more specifically, by strong and weak observations on quantum systems.

  The wavefunction can be understood as an avatar of the state-function in the context of QM. However, while the state-function can be constantly observable in the classical sense, the wavefunction, when subjected to an observation, collapses. When this happens, what was earlier a superposition of multiple eigenstates, metaphorical to physical realities, becomes resolved, in a manner of speaking, into one. This counter-intuitive principle was best summarized by Erwin Schrodinger in 1935 as a thought experiment titled…

  This aspect of observation, as is succinctly explained in the video, is what forces nature’s hand. Now, we pull in Werner Heisenberg and his notoriously annoying principle of uncertainty: if either of two conjugate parameters of a particle is measured, the value of the other parameter is altered. However, when Heisenberg formulated the principle heuristically in 1927, he also thankfully formulated a limit of uncertainty. If a measurement could be performed within the minuscule leeway offered by the constant limit, then the values of the conjugate parameters could be measured simultaneously without any instantaneous alterations. Such a measurement is called a “weak” measurement.

  Now, in the brain, if our ability to imagine could be ascribed – figuratively, at least – to our ability to “weakly” measure the properties of a quantum system via its wavefunction, then our brain would be able to comprehend different information-states and eventually arrive at one to act upon. By extension, I may not be implying that our brain could be capable of force-collapsing a wavefunction into a particular state… but what if I am? After all, the CTM does require inputs to be deterministic.

  

  By moving upward from the infraconscious domain of applicability of the CTM to the more complex cognitive functions, we are constantly teaching ourselves how to perform different kinds of tasks. By inculcating a vast and intricately interconnected network of simple memories and meanings, we are engendering the emergence of complexity and complex systems. In this teaching process, we also inculcate the notion of free-will, which is simply a heady combination of traditionalism and rationalism.

  While we could be, with the utmost conviction, dreaming up nonsensical images in our heads, those images could just as easily be the result of parsing different memories and meanings (that we already know), simulating them, “weakly” observing them, forcing successive collapses into reality according to our traditional preferences and current environmental stimuli, and then storing them as more memories accompanied by more semantic connotations.

  2012.11.13